Theorem 4ralbii

Description: Inference adding four restricted universal quantifiers to both sides of an equivalence. (Contributed by Scott Fenton, 28-Feb-2025)

	Ref	Expression
Hypothesis	4ralbii.1	⊢ ( 𝜑 ↔ 𝜓 )
Assertion	4ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		4ralbii.1	⊢ ( 𝜑 ↔ 𝜓 )
2	1	ralbii	⊢ ( ∀ 𝑤 ∈ 𝐷 𝜑 ↔ ∀ 𝑤 ∈ 𝐷 𝜓 )
3	2	3ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑤 ∈ 𝐷 𝜓 )